Research on hybrid reservoir scheduling optimization based on improved walrus optimization algorithm with coupling adaptive ε constraint and multi-strategy optimization

Reservoir flood control scheduling is a challenging optimization task, particularly due to the complexity of various constraints. This paper proposes an innovative algorithm design approach to address this challenge. Combining the basic walrus optimization algorithm with the adaptive ε-constraint method and introducing the SPM chaotic mapping for population initialization, spiral search strategy, and local enhancement search strategy based on Cauchy mutation and reverse learning significantly enhances the algorithm's optimization performance. On this basis, innovate an adaptive approach ε A New Algorithm for Constraints and Multi Strategy Optimization Improvement (ε-IWOA). To validate the performance of the ε-IWOA algorithm, 24 constrained optimization test functions are used to test its optimization capabilities and effectiveness in solving constrained optimization problems. Experimental results demonstrate that the ε-IWOA algorithm exhibits excellent optimization ability and stable performance. Taking the Taolinkou Reservoir, Daheiting Reservoir, and Panjiakou Reservoir in the middle and lower reaches of the Luanhe River Basin as a case study, this paper applies the ε-IWOA algorithm to practical reservoir scheduling problems by constructing a three-reservoir flood control scheduling system with Luanxian as the control point. A comparative analysis is conducted with the ε-WOA, ε-DE and ε-PSO (particle swarm optimization) algorithms.The experimental results indicate that ε-IWOA algorithm performs the best in optimization, with the occupied flood control capacity of the three reservoirs reaching 89.32%, 90.02%, and 80.95%, respectively. The control points in Luan County can reduce the peak by 49%.This provides a practical and effective solution method for reservoir optimization scheduling models. This study offers new ideas and solutions for flood control optimization scheduling of reservoir groups, contributing to the optimization and development of reservoir scheduling work.

algorithm's practical applications, this paper takes the Taolinkou Reservoir, Daheiting Reservoir, and Panjiakou Reservoir in the upper and middle reaches of the Luanhe River as the research objects.It establishes a flood control scheduling model for the mixed reservoir group.The ε-IWOA algorithm is used for solving and compared with ε-DE, ε-WOA and ε-PSO algorithms through experiments.The research results indicate that the algorithm exhibits excellent optimization performance and possesses good robustness, making it a practical and effective method for solving reservoir flood control scheduling models.
The structure of this paper is as follows: section "The optimal flood control scheduling model for reservoir groups" elaborates on the construction process of the flood control optimization scheduling model for reservoir groups, providing a solid theoretical foundation for subsequent scheduling research.Section "Adaptive ε-constrained IWOA algorithm" introduces the ε-IWOA algorithm and verifies its performance and effectiveness in practical applications through a series of tests.The innovation and practicality of this algorithm provide technical solid support for subsequent case analysis.In Section "Case analysis", the case analysis section combines the joint scheduling of the reservoir group in the specific research area to deeply explore the practical application effects of the model and algorithm.It discusses and analyzes the results in detail.Finally, in Section "Conclusion", we summarize the main research results of the paper and draw corresponding conclusions.

Objective function
The flood control system for a river basin reservoir group typically encompasses multiple components such as reservoirs, dams, river channels, flood diversion areas, and downstream control points.Appreciatively large river basins may also include flood storage and detention areas.However, in the discussion of this paper, we do not delve into any content related to flood storage and detention areas.The generalized network structure of the flood control system can be visually presented in Fig. 1.
The objective of flood control scheduling for a reservoir group is to minimize the flood peak flow in the reservoirs and mitigate flood disasters in downstream protected areas.During the flood scheduling process, each reservoir's initial water level for flood control is set at the flood limit level.To prepare for the next flood event, the reservoir water level should be reduced to the flood limit level by the end of the current flood operation.This paper aims to control the flood volume in the reservoirs as much as possible during the scheduling process while minimizing the flood peak flow at the downstream control point.Taking a typical flood process as an example, the objective function of the optimization model is designed to achieve these objectives. where: 3 represent the normalized maximum storage capacities of the three reservoirs after flood scheduling, in billions of m 3 ; Q ′ represents the standardized peak flow at the downstream control point, m 3 /s; ω 1 , ω 2 , ω 3 , ω 4 are the weight factors for objectives 1, 2, 3, and 4, respectively (the weight factors in this paper are referenced from the research results of Chen et al. 28 ).To eliminate the influence of different units, the original values can be normalized using Eqs.( 2) and (3): In the formula: V i is the maximum storage capacity of the i-th reservoir during flood control scheduling, in billions of m 3 ;V z,i is the total flood control capacity of the i-th reservoir, in billions of m 3 ;V l,i is the storage capacity www.nature.com/scientificreports/corresponding to the flood control limit water level of the i-th reservoir, in billions of m 3 ; Q is the peak flow rate at the downstream control point during flood control scheduling, m 3 /s; Q max is the maximum discharge flow rate at the downstream control point without disaster based on historical data, m 3 /s; V z,i − V l,i is the storage capacity between the flood control limit water level and the maximum water level of the i-th reservoir, in billions of m 3 , known as the maximum flood control storage capacity;V ′ i is the proportion of flood storage occupied during the scheduling process of the i-th reservoir; for the i-th reservoir,V , i the smaller the value of, the safer the reservoir is.Additionally, for the downstream control point,Q ′ the smaller the value, the safer the downstream area is.

Constraints
(1) Water balance constraints where V t+1,n is the capacity of the nth reservoir at the end of the period,10 8 m 3 ;V t,n is the initial nth reservoir capacity, 10 8 m 3 ;Q t+1,n is the inflow to the nth reservoir in period t,m 3 /s;q t+1,n is the nth relief flow in period t,m 3 /s; t is to calculate the step size.
(2) Water level constraints where:Z min,n , Z max,n , Z n are the flood limit level of the nth reservoir, the flood defense high water level, and the water level during the flood control and scheduling process, respectively,m.
(3) Flood flow limitation levels where q n is the flow from the nth reservoir;q max,n is the discharge capacity of the nth reservoir at each period, billion m 3 .
(4) Water level restrictions where Z end,n , Z e,n is the water level at the end of reservoir scheduling and the limit water level, respectively,m.

Fusion of SPM chaotic mapping
The traditional whale optimization algorithm exhibits significant randomness during the initialization process, leading to a decline in the whale's foraging ability.Based on this, a method based on SPM chaotic mapping is proposed, which can effectively enhance the diversity and randomness of the population, thus improving the algorithm's optimization and convergence speed.As evidenced by literature 29 , SPM chaotic mapping demonstrates strong search capabilities and enriches the diversity of the population during initialization.The formula is as follows: where n ∈ (0, 1), u ∈ (0, 1), the system is in a chaotic state, and r is a random number between 0 and 1.

Fusion spiral search strategy
Integrating the spiral search strategy 30 enables the whales to possess multiple search paths to better adjust their positions, thus enhancing the global search performance of the algorithm.The updated formula for the whale's position with the spiral exploration is as follows: where z is the spiral exploration factor, b is the spiral shape constant, and p denotes the path coefficient, a random number.
Incorporating Cauchy variation and reverse learning strategies Tizhoosh et al. 31 proposed an innovative opposite learning strategy in 2005.The core idea of this strategy is to find the corresponding opposite solutions based on the currently existing solutions through the opposite learning method.Subsequently, the optimal solutions among these opposite solutions are evaluated and compared to select and retain the more excellent ones.This approach provides a new solution for optimization problems and helps to enhance the performance and efficiency of the algorithm.To further improve the whale optimization www.nature.com/scientificreports/algorithm's ability to find optimal solutions, this paper introduced the concept of opposite learning.The mathematical representation is as follows: where x ′ best (t) is the reverse solution of the optimal solution for the t-th generation, Ub and lb are upper and lower bounds, respectively, and r is a 1 that follows the (0,1) standard uniform distribution × The random number matrix of d (d is the spatial dimension), b1 represents the information exchange control parameter, and the formula is as follows: By utilizing the Cauchy operator to update the objective, we can leverage its adjustment capabilities to avoid the algorithm getting trapped in local optima.The calculation formula is as follows: To further enhance the search capability of the optimization algorithm, this paper introduce a hybrid strategy that alternately executes the opposite learning strategy and the Cauchy operator perturbation strategy with a certain probability, thereby dynamically updating the target position.Through opposite learning, we can obtain opposite solutions, which greatly expands the search range of the algorithm and enhances its global search capability.Additionally, this paper set dynamically changing upper and lower bounds, ub and lb, respectively.Compared to strategies with fixed boundaries, this dynamic boundary approach is more conducive to the optimization process of the algorithm.
In the Cauchy mutation strategy, this paper utilize mutation operators to perform mutation operations on the current best position, generating new solutions.This strategy can effectively overcome the drawback of the algorithm easily getting trapped in local optima, thereby enhancing its local search capability.By alternately executing these two strategies, we can strike a balance between global and local searches, further improving the search efficiency and accuracy of the algorithm.The selection probability ps, which determines which strategy to choose for updating, is defined as follows: where is the adjustment factor, after many experiments, taking the value of 0.05 when the function optimization results are optimal.

Adaptive ε-constraint method
To efficiently explore and develop the search space while balancing population diversity and convergence, Takahama 32 proposed an adaptive ε-constraint method.By adopting an improved individual comparison criterion, the information of excellent infeasible individuals is fully utilized to enhance the exploration capability of the search region and effectively avoid local optima issues.Additionally, the use of an adaptive ε-parameter adjustment strategy balances the relationship between feasible and infeasible individuals, strengthening the algorithm's search efficiency and robustness.These improvements help the algorithm solve optimization problems more comprehensively and efficiently, enabling it to find global optimal solutions 33 .The improvements are as follows: (1) By optimizing the individual comparison criterion, having successfully enhanced the diversity of the population, ensuring that both feasible and infeasible regions evolve towards the global optimal solution.This improvement not only enriches the exploration range of the search space but also ensures the convergence of the algorithm, thereby improving the efficiency and accuracy of finding the global optimal solution.The criterion formula is as follows: where: G(x) is the magnitude of constraint violation, f(x) is the value of function fitness, and Ps is a random number on the interval [0.9, 1].
(2) When the feasible region is relatively narrow, the algorithm may generate a relatively large ε-value.This strategy aims to expand the search range, thereby increasing the possibility of the algorithm escaping from local optima.Conversely, when the feasible region is broader, the algorithm generates a smaller ε-value, which helps enhance the population's development capability and accelerate the convergence process.Based on this idea, (10) this paper propose an adaptive ε-adjustment strategy that optimizes the algorithm's performance by dynamically adjusting the ε-value.The formula is as follows: where Te is the number of truncated evolutionary iterations, N is the number of populations , and is the proportion of feasible individuals in the population.The value of Te is particularly important, as being too large may result in too many infeasible solutions during the iteration process, thereby affecting the convergence of the population.Being too small eliminates many infeasible individuals in the early stages of iteration, making it easy to fall into local optima.

Steps of the ε-IWOA algorithm
ε-IWOA algorithm flow with reference to Fig. 2 .The main steps are: a. Initialize parameters, including the maximum number of iterations T, population size N, boundary conditions (such as upper and lower bounds) a max ,a min the dimension of decision variables, and the truncated evolution iteration count.b.Utilize the SPM chaotic mapping in formula (6) to initialize the population and calculate the fitness value of each individual.c.Update the position of the whales based on the spiral search strategy in formula (7) to explore a broader search space.d.Combine the Cauchy mutation and opposite learning strategies in formulas (8) to (12) to enhance the whales' search capability during foraging, thereby seeking better solutions.e. Update the position of the new generation of whales and recalculate their fitness values.f.Compare individuals according to formula (13) and select individuals with better fitness through a certain selection mechanism.g.Check if the termination criteria for iteration are met, such as reaching the maximum number of iterations or satisfying a convergence criterion.If the conditions are met, proceed to step h; otherwise, return to step c to continue iterating.h.Determine the final best fitness value of the whales and the corresponding optimal individual.

ε-IWOA simulation experimental testing
To verify the effectiveness of the newly proposed ε-IWOA algorithm,this paper conducted a series of tests using 24 constrained optimization test functions.The results were then compared in detail with those obtained from the basic ε-WOA and ε-DE algorithms.In the experimental setup, this paper ensured that the population size for all three algorithms was set to 200, the maximum number of iterations was set to 20,000, the maximum number of function evaluations was limited to 500,000, the truncation evolution iteration number was set to 1000, and the tolerance for equation constraint violation was set to 0.0001.
To eliminate potential biases caused by the randomness of the algorithms, this paper ensured that each of the three algorithms was independently run 30 times for each test function.The experimental results showed that for each test problem, all three algorithms could find feasible solutions as the final optimal solutions in 30 independent runs.However, significant differences emerged in the quality of the optimal solutions obtained by different algorithms.To present this difference more visually, this paper compiled a statistical table (See Table 1 in the Appendix) showing the objective function values of the optimal solutions obtained by the three algorithms for a detailed comparative analysis, with bold font indicating the best performance.
The detailed data presented in the above table shows the significant effectiveness of the ε-constraint method in assisting the WOA algorithm in solving constrained optimization problems.The ε-IWOA algorithm achieved excellent optimization results in most tested functions by combining the improved WOA algorithm with the ε-constraint method.Notably, the ε-IWOA algorithm found the globally optimal solutions of the tasks and exhibited a minor standard deviation, indicating more stable results.Furthermore, the average values obtained by the ε-IWOA algorithm were lower than those of the other two comparison algorithms, demonstrating its excellent robustness.
For the more challenging functions g2 and g13, the ε-IWOA algorithm proposed in this paper exhibited significant advantages compared to the other two comparison algorithms.In terms of optimal values, the ε-IWOA algorithm found the optimal solutions successfully.It showed a minor standard deviation, fully demonstrating its superior performance in global search ability and optimization accuracy.On most of the tested functions, the ε-IWOA algorithm exhibited high stability in 30 solution experiments.Except for functions g20 and g22, the algorithm successfully found the optimal solutions that satisfied the constraint conditions.This achievement was attributed to the introduction various innovative strategies in the basic WOA algorithm, including population initialization methods based on SPM chaotic mapping, spiral search strategies, and local intensification search strategies based on Cauchy mutation and reverse learning.The effective coupling of these strategies with the adaptive ε-constraint method jointly enhanced the optimization ability of the algorithm, achieving optimal levels in both solution accuracy and stability.

Overview of the research area
The Luanhe River Basin is located in the northeastern part of the North China Plain, with geographical coordinates spanning from 115° 34′ E to 119° 50′ E and from 39° 10′ N to 42° 30′ N. It boasts a vast drainage area of 44,800 km 2 .The mainstream of the Luanhe River runs southeast, traversing the Yanshan Mountains and the Jidong Plain, with a total length of 888 km.The climate in this region is characterized as a temperate continental monsoon climate with distinct seasonal features.The spring and autumn seasons are dry with little rainfall, while the winter is cold and arid.In contrast, the summer is hot and rainy.There is significant unevenness in precipitation distribution, with notable monthly variations.Specifically, summer rainfall ranges from 200 to 550 mm, accounting for 66% to 76% of the annual precipitation.July and August are the most rainy months, with their combined rainfall contributing 50% to 65% of the annual total.Due to the concentrated rainfall, the runoff of the Luanhe River fluctuates significantly throughout the year.Especially during the flood season in July and August, the inflow is particularly abundant, accounting for more than half of the annual total.Conversely, the inflow in January and February is relatively scarce, less than one-tenth of the annual total.Figure 3 depicts the water system map of the Luanhe River Basin, clearly illustrating the distribution and flow directions of the river system in this region.
In this study, we have selected three key reservoirs in the middle and lower reaches of the Luanhe River for in-depth investigation: the Taolinkou Reservoir, the Daheiting Reservoir, and the Panjiakou Reservoir.Table 1 provides a detailed overview of the key characteristics of these reservoirs.The Panjiakou Reservoir plays a pivotal role in the flood control and disaster reduction system of the Luanhe River.It not only can regulate floods but also effectively develops water resources and mitigates flood peaks when necessary, thereby reducing the impact of disasters.The Taolinkou Reservoir, located in the lower reaches of the Luanhe River near Tangshan, is a crucial agricultural water conservancy project.The high-quality surface water it provides is indispensable for soil improvement in the irrigation area downstream of the Luanhe River and for optimizing the growth environment of rice paddies.Meanwhile, the Daheiting Reservoir serves multiple functions, including coordinating with the Panjiakou Reservoir in water transfer and water level elevation, as well as trans-basin water transfer and retaining water from a wider area, providing strong support for regional water resource management.

Flood evolution
The Xin'anjiang model, created by Zhao 34 , has been widely and successfully applied in the field of flood forecasting 35,36 .The core strategy of this model is to refine the complex watershed system into multiple subunits.By meticulously calculating the runoff and confluence conditions of each sub-unit, the flooding process at the outlet of each sub-unit is accurately depicted.For detailed information on the theoretical background and parameter optimization of the Xin'anjiang model, please refer to Zhao's work 36 .
Taking Luanxian as a critical control point in the lower reaches of the Luanhe River, it effectively regulates all floods within this section.These floods are mainly triggered by high-intensity, long-duration, and widespread rainfall.The flooding process primarily consists of five key stages: floods upstream of the Panjiakou Reservoir, floods from Panjiakou to the Daheiting Reservoir, floods from Daheiting to Luanxian, floods upstream of the  During the flood evolution process in the Luanhe River Basin, as illustrated in Fig. 4 (Flood Process Generalized Diagram), this paper have several critical flood flows to consider.Assuming that the inflow to the Panjiakou Reservoir is Q1, the flood flow between Panjiakou and the Daheiting Reservoir is Q2, the flood flow between Daheiting and the control point of Luanxian is Q3, the flood flow upstream of the Taolinkou Reservoir is Q4, and the flood flow between Taolinkou and Luanxian is Q5.After regulation by the Panjiakou Reservoir, its outflow becomes q1.The inflow to the Daheiting Reservoir is the sum of the flow Q2 from the Panjiakou to Daheiting interval and the outflow q1 from the Panjiakou Reservoir.After regulation by the Daheiting Reservoir, its outflow becomes q2, which transforms into the flow q4 after the flood evolution process.Similarly, the outflow from the Taolinkou Reservoir is q3, which transforms into the flow q5 after flood evolution.This process reveals the complex dynamics of flood generation and evolution in the Luanhe River Basin.
In this study, the Xin'anjiang model was utilized to conduct a thorough and detailed flood process forecast for the five regions previously delineated.Figure 5 specifically illustrates the rainfall patterns in these five subregions during a typical flood event.This flood forecast spans 144 h, with precise analysis conducted at a threehour calculation cycle.
The downstream water flows from Daheiting Reservoir, and Taolinkou Reservoir continues to move towards Luan County along the river.Regarding the process of flood evolution, this paper have adopted the linear Muskingum flood routing method for calculation, with specific details as follows:  www.nature.com/scientificreports/ In the Muskingum flood routing model, S(t) represents the storage capacity of the river at time t; K represents the storage time constant of the river segment, and x represents the weight factor.I(t) and O(t) denote the inflow and outflow rates at time t, respectively.The parameters of the Muskingum model for the study area have been determined.The combined process of the three reservoirs is described as follows: The inflow rate to Panjiakou Reservoir is denoted as Q1, and its outflow rate is q1.The inflow rate to Daheiting Reservoir consists of two parts: the local inflow rate Q2 and the outflow rate q1 from Panjiakou Reservoir.Since Panjiankou Reservoir is relatively close to Daheiting Reservoir, q1 does not require additional adjustment calculations and is directly considered a component of the inflow rate to Daheiting Reservoir.The outflow rate from Daheiting Reservoir is q2.
The inflow rate to Taolinkou Reservoir is Q4, and its outflow rate is q3.At the Luan County control point, the flow rate Q consists of the following four components: the outflow rate q4 from Daheiting Reservoir, the outflow rate q5 from Taolinkou Reservoir, the inflow rate Q3 between Daheiting Reservoir and the Luan County control point, and the inflow rate Q5 between Taolinkou Reservoir and the Luan County control point.

Analysis of scheduling results
The core of reservoir flood control scheduling lies in precisely implementing flood discharge, flood detention, and flood storage operations.Through these scientific measures, we can effectively reduce the property losses caused by flood disasters and promote the stable and efficient development of reservoir scheduling work.To further optimize the joint scheduling strategy for the reservoir group of Taolinkou Reservoir, Daheiting Reservoir, and Panjiakou Reservoir in the Luanhe River Basin, we introduced the ε-IWOA algorithm.In the algorithm parameter settings, we set the population size POP to 200, the maximum number of iterations T to 200,000, and the truncation iteration count Te to 1000.To ensure the accuracy of the solution, the paper conducted 20 independent operations, resulting in the optimal joint flood control optimization scheduling scheme, as detailed in Table 2. See Figs.1-3 in the Appendix visually present these three reservoirs' flood control scheduling details.The optimized joint scheduling scheme after the ε-IWOA algorithm adjustment shows the original combined flow process, the flood regulation discharge of each reservoir, the combined flow process, and the incoming water situation at the Luanxian control point, as illustrated in Fig. 6.From the chart data, it is evident that the peak

Figure 3 .
Figure 3. Water System Map of the Luanhe River Basin (Produced by ArcMap 10.8).

Figure 4 .
Figure 4. Evolution of Probable Floods in Luan River Basin.

Figure 5 .
Figure 5. Flood forecast for five zones during a typical rainfall event.

Table 1 .
Parameters characterizing the reservoir.

Panjiakou reservoir in Hebei province Dahaiting reservoir Touring the Taolinkou reservoir
Taolinkou Reservoir, and floods from Taolinkou to Luanxian.Characterized by high flood peaks, short durations, and high flow velocities, these five types of floods often cause devastating disasters in downstream areas.

Table 2 .
Statistics of optimized scheduling results for joint flood control.